Lycée Des Arts
Name: . . . . . . . . . .
AI-
Mathematics
GCD & LCM
8th-Grade
A.S-1
Notion about numbers:
Sort the following numbers in two sets and assign a name for each set:
2,3, 1, 5, 0,11,7, 3 & 6
II- What is:
a. A natural number? ……………………………………………………………………….
b. An integer? ………………………………………………………………………………
Restrictions: In this chapter our only concern is non-zero natural numbers
B- GCD of two or more non-zero natural numbers:
Note: If a & b are any two non-zero natural numbers, then GCD a; b  , is the short hand for
greatest common divisor of a & b .
Part-1: Reviewing GCD:
1-
Find the set of natural divisors of:
D4  {..................................}
a. 8 & 4 : D8  {..................................}
D27  {..................................}
b. 9 & 27 : D9  {..................................}
2- Determine:
a. The GCD 8;4  …………………………………
b. The GCD 9;27  ……………………………….
3- What can you say about:
a. 8 & 4 ? …………………………………………………………
b. 9 & 27 ? ……………………………………..…………………
c. a & 3a ? ……………………………………..………………… (condition: a  0 )
4- Deduce the GCD a;3a  ……………………………
 Complete the following: If a is a …………………... of b , then GCD a; b   .………...
Rule-1
8th Grade.
The GCD of any two non-zero multiples is the greater.
Mathematics-GCD & LCM.
Page 1 of 5
5-
Find the set of divisors of:
D7  {..................................}
a. 6 & 7 : D6  {..................................}
D4  {..................................}
b. 3 & 4 : D3  {..................................}
Determine:
a. The GCD 6;7  …………………………………
b. The GCD 3;4  ……………………………….
What can you say about:
a. 6 & 7 ? …………………………………………………………
b. 3 & 4 ? ……………………………………..…………………
c. a & a  1 ? ……………………………………..…………………
Deduce the GCD a; a  1 …………………………………… (condition: a  0 )
6-
7-
8-
 What do you conclude? ........................................................................................................
Rule-2
The GCD of any two non-zero consecutives is their product.
9-
Find the set of divisors of:
D7  {..................................}
a. 2 & 7 : D2  {..................................}
D19  {..................................}
b. 11&19 : D11  {..................................}
10- Determine:
a. The GCD 2;7  …………………………………
b. The GCD 11;19 ……………………………….
11- What can you say about:
a. 2 & 7 ? …………………………………………………………
b. 11&19 ? ……………………………………..………………….

What do you conclude? ..........................................................................................................
Note that:


If the GCDa; b   1 , then a & b are called relatively prime number or coprime.
Is the converse true? Justify. …………………………………………………………..
Rule-3
8th Grade.
If GCD of any two non-zero numbers is 1 then they are coprime.
Mathematics-GCD & LCM.
Page 2 of 5
Part-2: Determination of GCD of two or more non-zero natural number:
12- Determine the GCD of each of the following pairs, using prime factorization:
GCD of
One by one prime factors
At the same time
90 & 243
576 & 420
150,240 & 320
490,252 & 660
C- LCM of two or more non-zero natural numbers:
Note: If a & b are any two non-zero natural numbers, then LCM a; b  , is the short hand for
Least common multiple of a & b .
Part-1: Reviewing LCM:
1)
Find the set of multiples of: (list first six multiples only)
M 6  {..................................}
a. 3 & 6 : M 3  {..................................}
M 27  {..................................}
b. 15 & 60 : M 9  {..................................}
2)
Determine:
c. The LCM 3;6  …………………………………
d. The LCM 15;60 ……………………………….
8th Grade.
Mathematics-GCD & LCM.
Page 3 of 5
3)
What can you say about:
i- 3 & 6 ? ……………………………………..…………………….
ii- 15 & 60 ? ……………………………………..…………………
iii- 2a & 6a ? ……………………………………..…………………
4)
Deduce the LCM 2a;6a  ………………………………………….
 Complete the following conclusion: If a
Rule-4
is a ……………of b , then LCM a; b   ………...
The LCM of any two non-zero multiples is the least of them.
5)
Find the set of multiples of: (list first seven multiples only)
M 6  {..................................}
M 5  {..................................}
i- 5 & 6 :
M 4  {..................................}
M 3  {..................................}
ii- 3 & 4 :
6)
Determine:
i- The LCM 5;6  ………………………………….
ii- The LCM 3;4  ………………………………….
7)
What can you say about:
i- 5 & 6 ? …………………………………………………………
ii- 3 & 4 ? ……………………………………..…………………
iii- a & a  1 ? ……………………………………..…………………
8)
Deduce the LCM a; a  1 ……………………………
 What do you conclude? ........................................................................................................
Rule-5
9)
The LCM of any two consecutive numbers is their product.
Find the set of multiples of: (list first eight multiples only)
M 5  {......................................}
i- 3 & 5 : M 3  {......................................}
M 7  {......................................}
ii- 7 & 5 : M 5  {......................................}
10) Determine:
i- The LCM 3;5 …………………………………
ii- The LCM 7;5 ……………………………….
8th Grade.
Mathematics-GCD & LCM.
Page 4 of 5
11) What can you say about:
i- 3 & 5 ? …………………………………………………………
ii- 5 & 7 ? ……………………………………..………………….
 What do you conclude? ..........................................................................................................
Rule-6
The LCM of any two prime numbers is their product.
Part-2: Determination of LCM of two or more non-zero natural number:
12) Determine using prime factorization the:
LCM of
One by one prime factors
At the same time
90 & 60
150,240 & 320
490,252 & 660
8th Grade.
Mathematics-GCD & LCM.
Page 5 of 5